Description Usage Arguments Details Value Author(s) References See Also Examples
This function generates a posterior density sample for a Dirichlet process mixture of normals model for interval-censored data.
1 | DPMdencens(left,right,ngrid=100,grid=NULL,prior,mcmc,state,status)
|
left |
a vector or matrix giving the lower limit for each response variable. Note that the responses are defined on the entire real line and that unknown limits should be indicated by NA. |
right |
a vector or matrix giving the upper limit for each response variable. Note that the responses are defined on the entire real line and that unknown limits should be indicated by NA. |
ngrid |
number of grid points where the density estimate is evaluated. The default value is 100. |
grid |
matrix of dimension ngrid*nvar of grid points where the density estimate is evaluated. The default value is NULL and the grid is chosen according to the range of the interval limits. |
prior |
a list giving the prior information. The list includes the following
parameter: |
mcmc |
a list giving the MCMC parameters. The list must include
the following integers: |
state |
a list giving the current value of the parameters. This list is used if the current analysis is the continuation of a previous analysis. |
status |
a logical variable indicating whether this run is new ( |
This generic function fits a Dirichlet process mixture of normal model for density estimation (Escobar and West, 1995) based on interval-censored data:
yij in [lij,uij),i=1,…,n, j=1,…,m,
yi | mui, Sigmai ~ N(mui,Sigmai), i=1,…,n,
(mui,Sigmai) | G ~ G,
G | alpha, G0 ~ DP(alpha G0),
where, yi=(yi1,…,yim), and the baseline distribution is the conjugate normal-inverted-Wishart distribution,
G0 = N(mu| m1, (1/k0) Sigma) IW (Sigma | nu1, psi1)
To complete the model specification, independent hyperpriors are assumed (optional),
alpha | a0, b0 ~ Gamma(a0,b0)
m1 | m2, s2 ~ N(m2,s2)
k0 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)
psi1 | nu2, psi2 ~ IW(nu2,psi2)
Note that the inverted-Wishart prior is parametrized such that if A ~ IWq(nu, psi) then E(A)= psiinv/(nu-q-1).
To let part of the baseline distribution fixed at a particular value, set the corresponding hyperparameters of the prior distributions to NULL in the hyperprior specification of the model.
Although the baseline distribution, G0, is a conjugate prior in
this model specification, an algorithm based on auxiliary parameters is adopted.
Specifically, the algorithm 8 with m=1 of Neal (2000) is considered in the DPMdencens
function.
Finally, note that this function can be used to fit the DPM of normals model for ordinal data
proposed by Kottas, Mueller and Quintana (2005). In this case, the arbitrary cut-off points must
be specified in left
and right
. Samples from the predictive distribution contained
in the (last columns) of the object randsave (please see below) can be used to obtain an estimate of
the cell probabilities.
An object of class DPMdencens
representing the DP mixture of normals
model fit. Generic functions such as print
, summary
, and plot
have methods to
show the results of the fit. The results include the baseline parameters, alpha
, and the
number of clusters.
The function DPrandom
can be used to extract the posterior mean of the
subject-specific means and covariance matrices.
The MCMC samples of the parameters and the errors in the model are stored in the object
thetasave
and randsave
, respectively. Both objects are included in the
list save.state
and are matrices which can be analyzed directly by functions
provided by the coda package.
The list state
in the output object contains the current value of the parameters
necessary to restart the analysis. If you want to specify different starting values
to run multiple chains set status=TRUE
and create the list state based on
this starting values. In this case the list state
must include the following objects:
ncluster |
an integer giving the number of clusters. |
muclus |
a matrix of dimension (nobservations+100)*(nvariables) giving the means of the clusters
(only the first |
sigmaclus |
a matrix of dimension (nobservations+100)*( (nvariables)*((nvariables)+1)/2) giving
the lower matrix of the covariance matrix of the clusters (only the first |
ss |
an interger vector defining to which of the |
alpha |
giving the value of the precision parameter. |
m1 |
giving the mean of the normal components of the baseline distribution. |
k0 |
giving the scale parameter of the normal part of the baseline distribution. |
psi1 |
giving the scale matrix of the inverted-Wishart part of the baseline distribution. |
y |
giving the matrix of imputed data points. |
Alejandro Jara <atjara@uc.cl>
Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.
Kottas, A., Mueller, P., Quintana, F. (2005). Nonparametric Bayesian modeling for multivariate ordinal data. Journal of Computational and Graphical Statistics, 14: 610-625.
Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.
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####################################
# Bivariate example:
# Censored data is artificially
# created
####################################
data(airquality)
attach(airquality)
ozone <- Ozone**(1/3)
radiation <- Solar.R
y <- na.omit(cbind(radiation,ozone))
# create censored-data
xxlim <- seq(0,300,50)
yylim <- seq(1.5,5.5,1)
left <- matrix(0,nrow=nrow(y),ncol=2)
right <- matrix(0,nrow=nrow(y),ncol=2)
for(i in 1:nrow(y))
{
left[i,1] <- NA
right[i,1] <- NA
if(y[i,1] < xxlim[1]) right[i,1] <- xxlim[1]
for(j in 1:length(xxlim))
{
if(y[i,1] >= xxlim[j]) left[i,1] <- xxlim[j]
if(y[i,1] >= xxlim[j]) right[i,1] <- xxlim[j+1]
}
left[i,2] <- NA
right[i,2] <- NA
if(y[i,2] < yylim[1]) right[i,2] <- yylim[1]
for(j in 1:length(yylim))
{
if(y[i,2] >= yylim[j]) left[i,2] <- yylim[j]
if(y[i,2] >= yylim[j]) right[i,2] <- yylim[j+1]
}
}
# Prior information
s2 <- matrix(c(10000,0,0,1),ncol=2)
m2 <- c(180,3)
psiinv2 <- diag(c(1/10000,1),2)
prior <- list(alpha=1,nu1=4,nu2=4,s2=s2,
m2=m2,psiinv2=psiinv2,tau1=0.01,tau2=0.01)
# Initial state
state <- NULL
# MCMC parameters
nburn <- 5000
nsave <- 5000
nskip <- 3
ndisplay <- 1000
mcmc <- list(nburn=nburn,
nsave=nsave,
nskip=nskip,
ndisplay=ndisplay)
# Fitting the model
fit1 <- DPMdencens(left=left,right=right,ngrid=100,
prior=prior,mcmc=mcmc,
state=state,status=TRUE)
fit1
summary(fit1)
# Plot the estimated density
plot(fit1)
# Extracting the univariate density estimates
cbind(fit1$grid[,1],fit1$funi[[1]])
cbind(fit1$grid[,2],fit1$funi[[2]])
# Extracting the bivariate density estimates
fit1$grid[,1]
fit1$grid[,2]
fit1$fbiv[[1]]
# Plot of the estimated density along with the
# true data points and censoring limits
contour(fit1$grid[,1],fit1$grid[,2],fit1$fbiv[[1]])
points(y)
abline(v=xxlim)
abline(h=yylim)
## End(Not run)
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